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Γεωμετρία Klein
Γεωμετρία Klein Kleinian Geometry Επιστημονικός Κλάδος της Γεωμετρίας. Ετυμολογία Το όνομα "Kleinian" σχετίζεται ετυμολογικά με το όνομα "Klein". Περιγραφή In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous space X'' together with a transitive action on ''X by a Lie group G'', which acts as the symmetry group of the geometry. For background and motivation see the article on the Erlangen program. Formal definition A '''Klein geometry' is a pair (G'', ''H) where G'' is a Lie group and ''H is a closed Lie subgroup of G'' such that the (left) coset space ''G/''H'' is connected. The group G'' is called the '''principal group' of the geometry and G''/''H is called the space of the geometry (or, by an abuse of terminology, simply the Klein geometry). The space X'' = ''G/''H'' of a Klein geometry is a smooth manifold of dimension :dim X'' = dim ''G − dim H''. There is a natural smooth left action of ''G on X'' given by : g\cdot(aH) = (ga)H. Clearly, this action is transitive (take ''a = 1), so that one may then regard X'' as a homogeneous space for the action of ''G. The stabilizer of the identity coset H'' ∈ ''X is precisely the group H''. Given any connected smooth manifold ''X and a smooth transitive action by a Lie group G'' on ''X, we can construct an associated Klein geometry (G'', ''H) by fixing a basepoint x''0 in ''X and letting H'' be the stabilizer subgroup of ''x''0 in ''G. The group H'' is necessarily a closed subgroup of ''G and X'' is naturally diffeomorphic to ''G/''H''. Two Klein geometries (G''1, ''H''1) and (''G''2, ''H''2) are '''geometrically isomorphic' if there is a Lie group isomorphism φ : G''1 → ''G''2 so that φ(''H''1) = ''H''2. In particular, if φ is conjugation by an element ''g ∈ G'', we see that (''G, H'') and (''G, gHg−1) are isomorphic. The Klein geometry associated to a homogeneous space X'' is then unique up to isomorphism (i.e. it is independent of the chosen basepoint ''x''0). Bundle description Given a Lie group ''G and closed subgroup H'', there is natural right action of ''H on G'' given by right multiplication. This action is both free and proper. The orbits are simply the left cosets of ''H in G''. One concludes that ''G has the structure of a smooth [[principal bundle|principal H''-bundle]] over the left coset space ''G/''H'': : H\to G\to G/H.\, Types of Klein geometries Effective geometries The action of G'' on ''X = G''/''H need not be effective. The kernel of a Klein geometry is defined to be the kernel of the action of G'' on ''X. It is given by : K = \{k \in G : g^{-1}kg \in H\;\;\forall g \in G\}. The kernel K'' may also be described as the core of ''H in G'' (i.e. the largest subgroup of ''H that is normal in G''). It is the group generated by all the normal subgroups of ''G that lie in H''. A Klein geometry is said to be '''effective' if K'' = 1 and '''locally effective' if K'' is discrete. If (''G, H'') is a Klein geometry with kernel ''K, then (G''/''K, H''/''K) is an effective Klein geometry canonically associated to (G'', ''H). Geometrically oriented geometries A Klein geometry (G'', ''H) is geometrically oriented if G'' is connected. (This does ''not imply that G''/''H is an oriented manifold). If H'' is connected it follows that ''G is also connected (this is because G''/''H is assumed to be connected, and G'' → ''G/''H'' is a fibration). Given any Klein geometry (G'', ''H), there is a geometrically oriented geometry canonically associated to (G'', ''H) with the same base space G''/''H. This is the geometry (G''0, ''G''0 ∩ ''H) where G''0 is the identity component of ''G. Note that G'' = ''G''0 ''H. Reductive geometries A Klein geometry (G'', ''H) is said to be reductive and G''/''H a reductive homogeneous space if the Lie algebra \mathfrak h of H'' has an ''H-invariant complement in \mathfrak g . Examples In the following table, there is a description of the classical geometries, modeled as Klein geometries. Υποσημειώσεις Εσωτερική Αρθρογραφία *Γεωμετρία *Ευκλείδεια Γεωμετρία *Υπερβολική Γεωμετρία *Ελλειπτική Γεωμετρία *Παραβολική Γεωμετρία *Προβολική Γεωμετρία Βιβλιογραφία * * Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *[ ] *[ ] Category: Επιστημονικοί Κλάδοι Γεωμετρίας